Observations of Underground Temperature. 27 



ing formula : — 



tan e ,.= 5.; M^CV + B, 8 )* 



(or for logarithmic calculation, 



M { =AjSec€ f ). 



The preceding investigation is sufficient as a solution of the 

 problem, to find a complex harmonic function expressing a 

 given arbitrary periodic function, when once we are assured that 

 the problem is possible ; and when we have this assurance, it 

 proves that the resolution is determinate, that is to say, that no 

 other complex harmonic function than the one we have found 

 can satisfy the conditions. Tor a thorough and most interest- 

 ing analysis of the subject, supplying all that is wanting to com- 

 plete the investigation, and giving admirable views of the pro- 

 blem from all sides, the reader is referred to Fourier's delightful 

 treatise. A concise and perfect synthetical investigation of the 

 harmonic expression of an arbitrary periodic function is to be 

 found in Poisson's Theorie Mathematique de la Chaleur, chap. vii. 



II. Periodic Variations of Terrestrial Temperature. 



7. If the whole surface of the earth were at each instant of 

 uniform temperature, and if this temperature were made to vary 

 as a perfectly periodic function of the time, the temperature at 

 any internal point must ultimately come to vary also as a periodic 

 function of the time, with the same period, whatever may have 

 been the initial distribution of temperature throughout the whole. 

 Fourier's principles show how the periodic variation of internal 

 temperature is to be conceived as following, with diminished 

 amplitude and retarded phase, from the varying temperature at 

 the surface supposed given : and by his formulae the precise law 

 according to which the amplitude would diminish and the phase 

 would be retarded, for points more and more remote from the 

 surface, if the figure were truly spherical and the substance 

 homogeneous, is determined. 



8. The largest application of this theory to the earth as a 

 whole is to the analysis of imaginable secular changes of tem- 

 perature, with at least thousands of millions of years for a period. 

 In such an application, it would be necessary to take into account 

 the spherical figure of the earth as a whole. Periodic variations 

 at the surface with any period less than a million* of years will, 



* A periodic variation of external temperature of one million years' period 

 would give variations of temperature within the earth sensible to one 

 thousand times greater depths than a similar variation of one year' s period. 

 Now the ordinary annual variation is reduced to ? V tn of its superficial 



